Finding the output voltage for op amp circuit

Question

I've got this circuit

and I want to find the output voltage \$U_A\$. We are given the values of the resistances, the input \$U_E\$ and the "supplying" Voltage \$U_V\$ (which isn't drawn here):

\$U_E = 10V\$; \$U_V = \pm 15V\$;

\$R_1 = R_2 = 3,9 k\Omega \$; \$R_3 = 8,2 k\Omega \$ ; \$R_4 = 2,2 k\Omega \$

We were supposed to assume that \$U_{E-} = 0\$ and \$ I_{E-} = I_{E+} = 0\$

Attempt at solution:

By KCL I know that: \$ I_1 = I_2 + I_3 \$ and also that \$ I_2 = I_4 \$

By KVL I know that: \$U_E - U_{R1} - U_{R2} = 0\$

and: \$U_E - U_{R_1} - U_{R3} - U_{R4} = 0\$

However, I don't know what happens to \$I_3\$. Is \$I_3\$ just \$I_4\$ or \$I_3 = ?? + I_4\$ and how all of this relates to \$U_A\$ (unless \$U_A = U_{R4}\$ but I am not sure).

Answer 1

Here's how I'd do it: -

Once you have found the voltage at this junction (call it Vx) the problem is easy: -

\$U_A = V_x\frac{-R_4}{R_2}\$

How to find Vx?

Maybe convert Ue and Ua to current sources like this: -

Then, the net current flowing through the parallel combination of R1, R2 and R3 is: -

\$\dfrac{U_E}{R_1} - \dfrac{U_A}{R_3}\$

That current multiplied by R1||R2||R3 is Vx

Can you take it from here?

Answer 2

Here's what you should do: Write node equation for node e and A as below:

• e: (e-A)/R3 + e/R2 + (e-E)/R1 = 0
• A: A/R4 + (A-e)/R3 = 0

The variable A stands for the potential U_A. Now you can solve for potential at node A.

Answer 3

Here comes an alternative method, which seems to be simpler because the whole process is divided in two parts:

1) Ground the opamp output node and find the voltage at the inverting opamp node (forward function Hf) caused by the input VE (see the first diagram).

2) Ground the input node and find the voltage at the inverting opamp node (return functiin Hr) caused by the opamps output voltage VA (see the second diagram).

3.) The ratio (-Hf/Hr) gives the wanted overall transfer function. This simply results from the well-known feedback formula (H. Black) for infinite gain.

^{simulate this circuit – Schematic created using CircuitLab}